Do the electric and magnetic components of an electromagnetic wave really generate each other? Frequently when EM waves are taught, it is said that the change in electric field causes a change in the magnetic field, which then causes a change in the electric field, and so on and so forth. 
But from my understanding of basic electromagnetics, it is not necessarily a changing electric field that creates a change in magnetic field, but instead an accelerating charge. and it is this charge that creates both the electric and magnetic field. Again, from my understanding, a changing magnetic field is not generated by a changing electric field, but instead just happens to always be present perpendicular to a changing electric field due to the laws of electromagnetism. 
Am I wrong? Or is the "mutual generation" concept between the electric and magnetic components of an EM wave an actual thing? 
 A: Do the electric and magnetic components of an electromagnetic wave really generate each other?
No they don't. Like Andrea said, they're two "aspects" of the same thing. And like you said, it's an electromagnetic wave. See the wiki article for  electromagnetic radiation where you can read that "the curl operator on one side of these equations results in first-order spatial derivatives of the wave solution, while the time-derivative on the other side of the equations, which gives the other field, is first order in time". One's the spatial derivative, the other's the time derivative. If it was a water wave and you were in a canoe, the tilt of your canoe is E and the rate of change of tilt is B. 
Frequently when EM waves are taught, it is said that the change in electric field causes a change in the magnetic field, which then causes a change in the electric field, and so on and so forth. 
Yes, and the people who say this tend to say this is why electromagnetic waves don't need a medium. But they do need a medium. Space is the medium. It isn't a medium like water or the ground, but it isn't nothing. See LIGO and note that they're trying to detect a length-change in the arms of the interferometer. That's essentially space waving. Also see this where Robert B Laughlin talks about quantum vacuum and likens space to window glass rather than Newtonian emptiness. 
and it is this charge that creates both the electric and magnetic field
IMHO you should avoid charge for now and stick to electromagnetic waves. 
Is the "mutual generation" concept between the electric and magnetic components of an EM wave an actual thing?
No it isn't. Check out this Wikipedia article about Jefimenko's equations: 
There is a widespread interpretation of Maxwell's equations indicating that spatially varying electric and magnetic fields can cause each other to change in time, thus giving rise to a propagating electromagnetic wave (electromagnetism). However, Jefimenko's equations show an alternative point of view. Jefimenko says, "...neither Maxwell's equations nor their solutions indicate an existence of causal links between electric and magnetic fields. Therefore, we must conclude that an electromagnetic field is a dual entity always having an electric and a magnetic component simultaneously created by their common sources: time-variable electric charges and currents".
A: This  plane polarized  wave from wikipedia may help


Electromagnetic waves can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. This 3D animation shows a plane linearly polarized wave propagating from left to right. Note that the electric and magnetic fields in such a wave are in-phase with each other, reaching minima and maxima together.

It is plotting the mathematical functions, solutions of Maxwell's equations, that describe a propagating wave.
$$\text{electric field}: E = E_m\sin(kx - \omega t) \\\\\\\ \text{magnetic field}: B = B_m \sin(kx - \omega t) $$

To be consistent with Maxwell's equations, these solutions must be related by

$$\frac{E_m}{B_m} = c$$
$E$ and $B$ are perpendicular to each other and in synchrony growing and diminishing as the wave propagates. The statement

the change in electric field causes a change in the magnetic field, which then causes a change in the electric field, and so on and so forth

refers to this graphic, where electric and magnetic fields increase and decrease in synchrony. 
The acceleration of charged particles generates electromagnetic waves and then the waves become independent of the source of the electric field that generated it and propagate according to the equations.
This is the classical framework. 
In the quantum mechanical framework the wave is built up by an enormous number of photons generated by the accelerated  electrons in the antenna , but that is another story.
A: 
"a changing magnetic field is not generated by a changing electric field, but instead just happens to always be present perpendicular to a changing electric field due to the laws of electromagnetism." 

So ... it is due to but not caused by. What is the difference? 
Short answer: it is not only "a thing" it is a correct thing.
This is much more clearly stated in the differential form of Maxwell's equations:
$$ \nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t} $$
$$ \nabla \times \mathbf{B} = \mu_0 \left( \mathbf{j} + \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}\right) \,.$$
And while magnetic fields can be cause by either current or changing electric fields, light will happily travel in charge-free regions of space, so there is no question at all about which term is at work in electromagnetic waves.
A: Maxwell's equations in vacuum are:
$$\nabla\cdot\mathbf{E} = 0$$
$$\nabla\cdot\mathbf{B} = 0$$
$$\nabla\times\mathbf{E} = -\frac{\partial\mathbf{B}}{\partial t}$$
$$\nabla\times\mathbf{B} = \frac{1}{c^2}\frac{\partial\mathbf{E}}{\partial t}$$
It's the last two of these that give rise to the interpretation that a changing magnetic field generates an electric field and vice versa. But you could always take the curl of both these equations, use the other two, and get simple, straightforward wave equations separately for $\mathbf{E}$ and $\mathbf{B}$:
$$\nabla^2\mathbf{E} - \frac{1}{c^2}\frac{\partial^2 \mathbf{E}}{\partial t^2} = 0$$
$$\nabla^2\mathbf{B} - \frac{1}{c^2}\frac{\partial^2 \mathbf{B}}{\partial t^2} = 0$$
While these wave equations mean that $\mathbf{E}$ and $\mathbf{B}$ propagate as ordinary waves, they are still constrained by the original relations; we haven't magically made that dependence disappear.
So, while it is correct to say that changing magnetic and electric fields generate each other (simultaneously), it is also correct to say that each propagates as a wave of its own right, but is everywhere related to the other by the original Maxwell equations.
Indeed, when a charge is accelerating, it does generate both a changing electric and magnetic field at its position. You can then say that they both propagate out as ordinary waves, but (repeat) are always constrained by Maxwell's equations to be interdependent.
A: As you say, a changing magnetic field is always associated with a changing electric field, and in fact in relativity they are finally revealed to be the same field. So at this level it cannot be said that the one field generates the other, as they are merely two aspects of the same object.
But maybe you still want to look at it from the perspective of "naive EM", and see what sense one could make of the statement that one field generates the other.
Now, if you look at the fields of a
 plane wave at a fixed point in space, you'll see that they oscillate in sync, and both reach maximum and $0$ intensity at the same time. In fact what you could say is that increasing $E$ field is trying to reduce the $B$ field and the increasing $B$ field is trying to reduce the $E$ field. You could follow the appropriate equations and you'll see this is analogous to the equations of a vibrating membrane.
But the point is, that in fact the fields try to mutually reduce each other rather than generate. In fact they do that so well, that there is an overshoot, and the cycle repeats.
This dynamics turns out to make the energy flow through that point in space, to the fields in the neighbouring space. In fact, at some moments there is no EM field at that point, which means there is no energy at that field.
I guess the best description would be that a moving charge generates a disturbance in the electromagnetic field, in the sense that there is an energy transfer from its kinetic energy to the EM energy. This disturbance than propagates through space and time.
A: Jefimenko believed that time varying electric and magnetic fields did not cause one another because Maxwell's equations only give the relationship between the fields at the same instant in time. He said the equation for the E field did not predict a future value for the B field - it just tells you what the B field is if you know the E field. This is unlike the calculation of E and B fields from earlier charge positions and movements using the retarded field equations which are patently a causal relationship.
However, taking Maxwell's equation:-  
$$\nabla\times\mathsf{E_{present}} = -\frac{\partial\mathsf{B_{present}}}{\partial t}$$
Re-write the rate of change of B as:-
$$\frac{B_{future}-B_{past}}{dt}$$
Substite this expression and re-arrange the terms in Maxwell's equation to give:-
$$B_{future}= B_{past}-dt{(\nabla\times\mathsf{E_{present}}})$$
Similarly using Maxwell's equation for curl B gives:-
$$E_{future}= E_{past}+c^2dt{(\nabla\times\mathsf{B_{present}}})$$
Written in this way the two Maxwell's equations are now seen to be predictive and causal. This is not just a mathematical trick as there is a numerical computational technique, known as the finite-difference time-domain (FDTD) method, which by alternately using each equation in a time stepping sequence can calculate exactly how the fields will propagate. It does not require any information about the charges or  currents which produced the initial fields, just the starting distribution of the electric and magnetic fields. An FDTD computer program provides a good demonstration of how the electric field generates a  magnetic field and the magnetic field generates an electric field. 
